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Question: Do fractals have projective duals? In the projective plane RP^2, the dual of a point is a line, and vice versa. Do fractals like "Burning Ship" and "Mandelbrot" have projective duals, where each point in the set is replaced with a line? What would these look like?
Great YouTube video of a deep zoom: https://www.youtube.com/watch?v=2S3lc2G3rWs
What's the trick to implement a zoom like this in software? I remember many years ago implementing a trivial program to display the mandelbrot fractal, but as you zoomed in you quickly ran out of precision even if you used 64bit double floats. Is it just using "bignums" behind the scenes or is there a trick to "reset" the exponent due to the fractal nature of the display? I always wondered if, thanks to the self-simi…
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Complex conjugation is non-holomorphic and yes, it's ugly. I still don't see why it being useful as a construction makes it a beautiful operation in analysis given that it isn't algebra. There are holomorphic split-complex functions and they aren't uninteresting at all - they are quite beautiful and complex, relating to wave equations: https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_f... They're not someth…
A holomorphic function over the split-complex numbers is the direct sum of two holomorphic functions over R. In that sense, it's trivial. You will be very disappointed once you see the split-complex Mandelbrot set. It will be an instructive exercise for you to see why it looks the way it does. My point about complex conjugation is illustrated in [1]. Linear algebra over the complex numbers does not limit itself to th…
Linear algebra does a lot of things, ie. transpose, that aren't algebraic. I think it's important we understand what is computational versus algebraic. In that vain, linear algebra proves to be immensely useful, but we will trip ourselves if we start thinking certain aspects of it are analytic.
Anyhow, point taken, that image hardly looks interesting.
I think we agree and like you said, are just talking about aesthetics. I agree that the utility of math doesn't end at algebra or analyticity.
These kind of fractal have actually 4 dimentional structure since c_x and c_y can also be parameters. I'd love to see their slice as 3D, but have yet to find a good way to visualize them... https://i.imgur.com/JRfLy6R.mp4
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Complex conjugation is non-holomorphic and yes, it's ugly. I still don't see why it being useful as a construction makes it a beautiful operation in analysis given that it isn't algebra. There are holomorphic split-complex functions and they aren't uninteresting at all - they are quite beautiful and complex, relating to wave equations: https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_f... They're not someth…
Here's the split-complex Mandelbrot (and Mandelbar for comparison): https://imgur.com/a/j2c7qkB Read my other reply.
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Just curious - do you have a script that finds related posts for each submission or do you rely on your memory, Dang?
Kind of a script, but not quite. Here's a pointer to past explanations: https://news.ycombinator.com/item?id=29370676 . Edit: or, to dereference all the way: https://news.ycombinator.com/item?id=26158300 .
(I had to click a lot of "here's a post where I explained it before" links that had "here's a post where I explained it before" before I found this one)
Incredibly creepy. What if one day we found a fractal that contains a message from God?
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Kind of a script, but not quite. Here's a pointer to past explanations: https://news.ycombinator.com/item?id=29370676 . Edit: or, to dereference all the way: https://news.ycombinator.com/item?id=26158300 .
If it helps any, here's the post you made that describes it most simply without needing to go 5 or 6 levels deep of self reference to the same question: https://news.ycombinator.com/item?id=26156901 (I had to click a lot of "here's a post where I explained it before" links that had "here's a post where I explained it before" before I found this one)
Earlier quoted context omitted.
Kind of a script, but not quite. Here's a pointer to past explanations: https://news.ycombinator.com/item?id=29370676 . Edit: or, to dereference all the way: https://news.ycombinator.com/item?id=26158300 .
If it helps any, here's the post you made that describes it most simply without needing to go 5 or 6 levels deep of self reference to the same question: https://news.ycombinator.com/item?id=26156901 (I had to click a lot of "here's a post where I explained it before" links that had "here's a post where I explained it before" before I found this one)
Earlier quoted context omitted.
Just curious - do you have a script that finds related posts for each submission or do you rely on your memory, Dang?
Kind of a script, but not quite. Here's a pointer to past explanations: https://news.ycombinator.com/item?id=29370676 . Edit: or, to dereference all the way: https://news.ycombinator.com/item?id=26158300 .